# Integration of Polynomials Times Double Step Function in Quadrilateral Domains for XFEM Analysis

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Numerical Methods in Fracture Mechanics

- FEM employs a structured mesh to discretise the domain into elements, resulting in efficient data representation, storage and manipulation and enabling the use of optimised algorithms and data structures to improve computational performance [49];
- FEM demonstrates excellent convergence properties. The accuracy of the solution improves as the mesh is refined. Convergence analysis plays a crucial role in assessing the reliability of numerical simulations [50];
- FEM generally provides higher accuracy for problems with smooth solutions. This advantage arises from the use of polynomial interpolation functions within each element, resulting in accurate approximations [51];
- XFEM provides a more stable solution and higher accuracy compared to many mesh-free methods in capturing stress and displacement fields near the crack tip by means of enrichment approximation functions [44];
- In XFEM, the crack geometry is implicitly represented within the finite elements, which reduces the dependency on the mesh density. This leads to a more efficient computational process and reduces the computational cost [12]. Mesh-free methods may require a large number of nodes or particles to accurately capture localised phenomena, such as cracks [44].

#### 2.2. Multiple Discontinuities Problems

- ${L}_{J}\subset I$ are the nodes to enrich for the j-th discontinuity, as such their support does not contain the ends of the discontinuity, and ${a}_{I,J}$ are the respective enriched degrees of freedom [26];
- ${K}_{J}\subset I$ are the nodes to enrich for the j-th discontinuity extremity, as such their support contains the ends of the discontinuity, and ${b}_{I,J}^{L},\phantom{\rule{0.222222em}{0ex}}L=1,...,4$ are the respective enriched degrees of freedom [26];
- ${J}_{J}\subset I$ are the nodes to enrich for the j-th junction, as such their support contains the j-th junction, and ${c}_{I,J}$ are the respective enriched degrees of freedom [26].

## 3. Materials and Methods

#### 3.1. Problem Statement

#### 3.2. Integration Algorithm for Single Discontinuity Problems

^{+}and $\mathsf{\Omega}$

^{−}(Figure 1).

**n**

^{+}in Figure 1 defines the positive portion of the element domain $\mathsf{\Omega}$ and points in the direction of $\mathsf{\Omega}$

^{+}. The coefficients a and b in (5) for the internal discontinuity line d are used to describe the components of the vector

**n**

^{+}, which is orthogonal to d.

^{+}(Figure 1) through the integral:

#### 3.3. Integration Algorithm for Double Discontinuity Problems

#### 3.4. Algorithm Description

- Calculating the signed distances (${D}_{i}$) in the global coordinate system between each discontinuity and each node of the integration domain;
- Writing the discontinuity coefficients (a, b and c) in the parent coordinate system as a function of ${D}_{i}$ by solving a linear equations system;
- Substituting the variables x and y in $\overline{q}\left(\mathit{x}\right)$ and $\overline{r}\left(\mathit{x}\right)$ by means of Equation (23), so that $q\left(\mathit{\xi}\right)$ and $r\left(\mathit{\xi}\right)$ are obtained in terms of the coefficients ${a}^{\prime}$, ${b}^{\prime}$ and ${c}^{\prime}$ dependent on ${D}_{i}$.

## 4. Results

#### 4.1. Library Architecture

- Primary data preparation:
- Individuation of the domain nodal coordinates in the global coordinate system;
- Individuation of the discontinuities coefficients in the global coordinate system;
- Selection of the domain portions to be evaluated.

- Isoparametric mapping onto the parent element domain and computation of the coefficient vector of the equivalent polynomial by means of the DD_Heqpol_coefficients subroutine.
- Quadrature by way of any chosen rule (i.e., (25)) in which the equivalent polynomial values at the quadrature points are provided by the function HeqPol and the Jacobian matrix determinant is given by the function detJ.

- The Jacobian of the transformation in (25) has to be constant;

#### 4.2. Numerical Examples

#### 4.2.1. Parallelogram Partitioned by Two Discontinuities Intersecting within the Element

- $q:\frac{7}{4}x-y-\frac{7}{2}=0$;
- $r:\frac{7}{4}x+y-\frac{21}{2}=0$.

`Input from file? (y/n): y``example_1.txt`

`\\ DOUBLE DISCONTINUITY EQP LIBRARY``\\ EXAMPLE 1: DISCONTINUITIES INTERSECTING INSIDE THE DOMAIN``$ElementType``\\ 21 : Quad``21``$Coords``\\ Set the coordinates for the element``\\ 1st col : x``\\ 2nd col : y``\\ Coordinates Scheme :``\\ Quad Element :``\\ 4-------------3``\\ | |``\\ | |``\\ | |``\\ | |``\\ | |``\\ 1-------------2``2.0 1.5``6.0 2.5``4.5 4.0``0.5 3.0``$NumOfDiscont``\\ Number of discontinuities crossing the element (1 or 2)``2``$DiscontCoefficients``\\ a,b,c coefficients for each discontinuity``\\ coefficients are separated by a blank``1.75 -1.0 -3.5``1.75 1.0 -10.5``$ElementPart``\\ In case of 2 discontinuities choose the element portion``\\ to integrate``\\ Part : A, B, C, D, all``\\ 3--------4``\\ | \A / |``\\ |B \/D |``\\ | / \ |``\\ |/ C \ |``\\ 1--------2``B`

#### 4.2.2. Parallelogram Partitioned by Two Discontinuities Intersecting Outside the Element

- $q:\frac{2}{9}\mathit{x}+\mathit{y}-\frac{9}{2}=0$;
- $r:\mathit{y}-2=0$.

#### 4.2.3. Outcomes

## 5. Discussion

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MDPI | Multidisciplinary Digital Publishing Institute |

FEM | Finite element method |

PUM | Partition of unity method |

CZM | Cohesive zone model |

BEM | Boundary element method |

SBFEM | Scaled boundary finite element method |

SPH | Smoothed particle hydrodynamics |

RKPM | Reproducing kernel particle method |

XFEM | Extended finite element method |

GFEM | Generalised finite element method |

EQP | Equivalent polynomials |

DD_EQP | Double discontinuity equivalent polynomials |

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**Figure 3.**Use of the auxiliary integration limit s to evaluate the equivalent polynomials ${\tilde{H}}_{i}\left(\mathit{x}\right)$. In the figure ${\tilde{H}}_{B}\left(\mathit{x}\right)={\tilde{H}}_{{q}^{+}}^{\left(s\right)}\left(\mathit{x}\right)-{\tilde{H}}_{{r}^{+}}^{\left(s\right)}\left(\mathit{x}\right)$. (

**a**) Integration domain evaluated by means of ${\tilde{H}}_{{q}^{+}}\left(\mathit{x}\right)$ with respect to the discontinuity q and the auxiliary integration limit s. (

**b**) Integration domain evaluated by means of ${\tilde{H}}_{{r}^{+}}\left(\mathit{x}\right)$ with respect to the discontinuity r and the auxiliary integration limit s.

**Figure 4.**Isoparametric mapping of a quadrangular element. (

**a**) Element configuration in the global coordinate system. (

**b**) Element configuration in the parent coordinate system.

**Figure 5.**Software illustrative examples. (

**a**) Example 1: parallelogram domain cut by two discontinuities intersecting inside the domain. (

**b**) Example 2: parallelogram domain cut by two discontinuities intersecting outside the domain.

**Table 1.**Integration domain, domain type, parent element domain and monomial basis included in the library.

Domain of Integration | Etype | Parent Domain | Monomial Basis |
---|---|---|---|

Parallelogram | 21 | $1,x,{x}^{2},y,xy,{y}^{2}$ |

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## Share and Cite

**MDPI and ACS Style**

Fichera, S.; Mariggiò, G.; Corrado, M.; Ventura, G.
Integration of Polynomials Times Double Step Function in Quadrilateral Domains for XFEM Analysis. *Algorithms* **2023**, *16*, 290.
https://doi.org/10.3390/a16060290

**AMA Style**

Fichera S, Mariggiò G, Corrado M, Ventura G.
Integration of Polynomials Times Double Step Function in Quadrilateral Domains for XFEM Analysis. *Algorithms*. 2023; 16(6):290.
https://doi.org/10.3390/a16060290

**Chicago/Turabian Style**

Fichera, Sebastiano, Gregorio Mariggiò, Mauro Corrado, and Giulio Ventura.
2023. "Integration of Polynomials Times Double Step Function in Quadrilateral Domains for XFEM Analysis" *Algorithms* 16, no. 6: 290.
https://doi.org/10.3390/a16060290